2006 Chapter Questions

MATHCOUNTS
®
2006
Chapter Competition
Sprint Round
Problems 1–30
Name
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This section of the competition consists of 30 problems.
You will have 40 minutes to complete all the problems.
You are not allowed to use calculators, books or other
aids during this round. Calculations may be done on
scratch paper. All answers must be complete, legible and
simpliﬁed to lowest terms. Record only ﬁnal answers in the
blanks in the right-hand column of the competition booklet.
If you complete the problems before time is called, use the
remaining time to check your answers.
In each written round of the competition, the required unit
for the answer is included in the answer blank. The plural
form of the unit is always used, even if the answer appears
to require the singular form of the unit. The unit provided
in the answer blank is the only form of the answer that will
be accepted.
Total Correct
Scorer’s Initials
Founding Sponsors
National Sponsors
National Society of
Professional Engineers
ADC Foundation
Raytheon Company
National Council of
Teachers of Mathematics
General Motors Foundation
Shell Oil Company
Lockheed Martin
Texas Instruments Incorporated
National Aeronautics and
Space Administration
3M Foundation
CNA Foundation
Northrop Grumman Foundation
Xerox Corporation
Copyright MATHCOUNTS, Inc. 2005. All rights reserved.
1. Two identical blue boxes together weigh
the same as three identical red boxes
together. Each red box weighs 10 ounces.
How much does one blue box weigh?
B
B
2. Three black chips have been placed on the
game board shown. What is the number in
the square where a fourth chip should be
placed so that it shares neither a row nor a
column with any of the existing chips?
R R R
1
2
3
4
5
6
7
8
9 10
11 12
ounces
1. ________________
2. ________________
13
3. What is the greatest number of cubes, with edge length
1 inch, that can
be placed into a
rectangular box
measuring 3 inches by
1
3 inches by 9 inches?
9
cubes
3. ________________
3
3
4. What fraction is equal to six times one-seventh? Express your
answer as a common fraction.
4. ________________
5. Krista put 1 cent into her new bank on a Sunday morning. On
Monday she put 2 cents into her bank. On Tuesday she put
4 cents into her bank, and she continued to double the amount
of money she put into her bank each day for two weeks. On
what day of the week did the total amount of money in her
bank ﬁrst exceed $2?
5. ________________
6. According to the table below, what is the median value of the
59 salaries paid to this company’s employees?
$
6. ________________
Position Title
President
Vice-President
Director
Associate Director
Administrative Specialist
# of Employees with
this Title
1
5
10
6
37
Salary
for Position
$130,000
$90,000
$75,000
$50,000
$23,000
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round
7. Each side of square ABCD
is doubled in length to form
square EFGH. The perimeter
of square EFGH is 40 cm.
What is the area of
square ABCD?
D
C
A
B
H
G
E
F
sq cm
7. ________________
8. New York and Denver are in different time zones. When it is
noon in New York, it is 10 a.m. in Denver. A train leaves New
York at 2 p.m. (New York time) and arrives in Denver 45 hours
later. What time is it in Denver when the train arrives?
a.m.
8. ________________
9. A survey of 400 patients at a hospital classiﬁed the patients
by gender and blood type, as shown in the table below. What
percent of the patients with type AB blood are male?
percent
9. ________________
Type A
45
55
100
Male
Female
TOTAL
Type B
60
40
100
Type O
80
100
180
Type AB
15
5
20
TOTAL
200
200
400
pounds
10. ________________
11. Roger has exactly one of each of the ﬁrst 22 states’ new U.S.
quarters. The quarters were released in the same order that the
states joined the union. The graph below shows the number of
states that joined the union in each decade. What fraction of
Roger’s 22 coins represents states that joined the union during
the decade 1780 through 1789? Express your answer as a
common fraction.
11. ________________
# of States that Joined
the Union
10. One type of cat food recommends that a cat have a daily
serving of 13 ounce of dry cat food per pound of body weight.
If a cat is fed 3 23 ounces of dry food a day according to these
recommendations, how many pounds does the cat weigh?
12
10
8
6
4
2
9
178
0178
69
29
89
49
09
09
-18 20-18 40-18 60-18 80-18 00-19
0
0
18
18
18
18
18
19
9
195
0195
Decades
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round
12. In the sequence 0, 1, 1, 3, 6, 9, 27, ..., the ﬁrst term is 0.
12. ________________
Subsequent terms are produced by alternately adding and
multiplying by each successive integer beginning with 1. For
instance, the second term is produced by adding
+1 ×1 +2 ×2
1 to the ﬁrst term; the third term is produced by
multiplying the second term by 1; the fourth term 0, 1, 1, 3, 6, 9, 27, ...
is produced by adding 2 to the third term; and so
on. What is the value of the ﬁrst term that is greater than 125?
13. Carolyn, Julie and Roberta share $77 in a ratio of 4:2:1,
respectively. How much money did Carolyn receive?
$
13. ________________
14. To weigh things on a balance scale, one or more objects are
placed on one pan and weights are placed onto the other pan
until the two pans are balanced. We see A is balanced by the
4 weights shown, and B is balanced by the 5 weights shown.
We have the following weights: 1, 1, 3, 3, 9, 9, 27 and 27.
What is the minimum number of these weights it would take to
balance the total weight of A plus B?
weights
14. ________________
A
A
3
9 3 1
B
9 3
9 3 1
B
?
15. If a = 2, b = 3 and c = 4, what is the numerical value of the
expression (b – c)2 + a (b + c) ?
15. ________________
16. The hexagon with the “R” is colored
red. Each hexagon is colored either
red, yellow or green, such that no two
hexagons with a common side are
colored the same color. In how many
different ways can the ﬁgure be colored?
ways
16. ________________
R
17. If one quart of paint is exactly enough for two coats of paint
on a 9-foot by 10-foot wall, how many quarts of paint are
needed to apply one coat of paint to a 10-foot by 12-foot wall?
Express your answer as a common fraction.
quarts
17. ________________
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round
sq units
18. ________________
19. The graph shows the total
distance Sam drove from
6 a.m to 11 a.m. How many
miles per hour is the car’s
average speed for the period
from 6 a.m. to 11 a.m.?
miles per hour
19. ________________
Total Driving Distance
Since 6 a.m. (miles)
18. Ten unit cubes are glued together as shown.
What is the surface area of the resulting solid?
160
120
80
40
0
6 7 8
9 10 11
Time of Day (a.m.)
20. A stock loses 10% of its value on Monday. On Tuesday it loses
20% of the value it had at the end of the day on Monday. What is
the overall percent loss in value from the beginning of Monday to
the end of Tuesday?
percent
20. ________________
21. The numbers 1, 2 and 3 are written in these nine unit squares.
• Each of the numbers appears three times, and
there is only one number placed in each of the
nine unit squares.
• Each number is in a unit square horizontally
or vertically adjacent to a unit square with the same number.
• The sum of the numbers in the leftmost column and the sum
of the numbers in the top row are each 7.
What is the sum of the numbers in the four shaded squares?
21. ________________
22. Five balls are numbered 1 through 5 and placed in a bowl. Josh
will randomly choose a ball from the bowl, look at its number and
then put it back into the bowl. Then Josh will again randomly
choose a ball from the bowl and look at its number. What is the
probability that the product of the two numbers will be even and
greater than 10? Express your answer as a common fraction.
22. ________________
23. If a certain negative number is multiplied by six, the result is the
same as 20 less than the original number. What is the value of the
original number?
23. ________________
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round
24. If 2004 is split after the third digit into a three-digit integer
and a one-digit integer, then the two integers, 200 and 4, have
a common factor greater than one. The years 2005 and 2006
each have this same property, too. What is the ﬁrst
odd-numbered year after 2006 that has this property?
24. ________________
25. Two consecutive even numbers are each squared. The
difference of the squares is 60. What is the sum of the original
two numbers?
25. ________________
26. What is the area of the pentagon shown here
with sides of length 15, 20, 27, 24 and 20 units?
15
20
square units
26. ________________
27
20
24
27. At the beginning of my bike ride I feel good, so I can travel
20 miles per hour. Later, I get tired and travel only 12 miles
per hour. If I travel a total of 122 miles in a total time of
8 hours, for how many hours did I feel good? Express your
answer as a common fraction.
hours
27. ________________
28. When reading a book, Charlie made a list by writing down the
page number of the last page he ﬁnished reading at the end of
each day. (He always ﬁnished reading a page that he started.)
His mom thought his list indicated the amount of pages he
had read on each day. At the end of the 8th day of reading, she
added the numbers on his list and thought Charlie had read
432 pages. If Charlie started reading the book on page one,
and he read the same amount of pages each day of this eightday period, how many pages did he actually read by the end of
the 8th day?
pages
28. ________________
COU
29. Suppose that each distinct letter in the
+ NT S
equation MATH = COU + NTS is replaced
MA T H
by a different digit chosen from 1 through 9
in such a way that the resulting equation is
true. If H = 4, what is the value of the greater of C and N?
29. ________________
1
30. One gear turns 33 3 times in a minute. Another gear turns
45 times in a minute. Initially, a mark on each gear is pointing
due north. After how many seconds will the two gears next
have both their marks pointing due north?
seconds
30. ________________
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Sprint Round
MATHCOUNTS
®
2006
Chapter Competition
Target Round
Problems 1 and 2
Name
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This section of the competition consists of eight problems,
which will be presented in pairs. Work on one pair of
problems will be completed and answers will be collected
before the next pair is distributed. The time limit for each
pair of problems is six minutes. The ﬁrst pair of problems
is on the other side of this sheet. When told to do so,
turn the page over and begin working. Record only ﬁnal
answers in the designated blanks on the problem sheet.
All answers must be complete, legible and simpliﬁed to
lowest terms. This round assumes the use of calculators,
and calculations may also be done on scratch paper, but
no other aids are allowed. If you complete the problems
before time is called, use the time remaining to check your
answers.
Total Correct
Scorer’s Initials
Founding Sponsors
National Sponsors
National Society of
Professional Engineers
ADC Foundation
Raytheon Company
National Council of
Teachers of Mathematics
General Motors Foundation
Shell Oil Company
Lockheed Martin
Texas Instruments Incorporated
National Aeronautics and
Space Administration
3M Foundation
CNA Foundation
Northrop Grumman Foundation
Xerox Corporation
Copyright MATHCOUNTS, Inc. 2005. All rights reserved.
1. Brass is an alloy created using 80% copper and 20% zinc. If
Henri’s brass trumpet contains 48 ounces of copper, how many
ounces of zinc are in the trumpet?
ounces
1. ________________
2. In the maze below, a player may only move toward the right.
At each junction the player chooses a path that includes an
operation (+ or –) and then a number. The player keeps a
running total of her score throughout her journey, starting with
7, and she may only choose among paths that keep her score
between 2 and 14 points at all times. After performing four
operations and ending at the 5, what is the lowest score with
which she can ﬁnish?
points
2. ________________
3
2
Start
with
7
4
5
3
2
4
3
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round
MATHCOUNTS
2006
Chapter Competition
Target Round
Problems 3 and 4
Name
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
Total Correct
Scorer’s Initials
Founding Sponsors
National Sponsors
National Society of
Professional Engineers
ADC Foundation
Raytheon Company
National Council of
Teachers of Mathematics
General Motors Foundation
Shell Oil Company
Lockheed Martin
Texas Instruments Incorporated
National Aeronautics and
Space Administration
3M Foundation
CNA Foundation
Northrop Grumman Foundation
Xerox Corporation
Copyright MATHCOUNTS, Inc. 2005. All rights reserved.
®
3. In the ﬁgure below, side AE of rectangle ABDE is parallel to
the x-axis, and side BD contains the point C. The vertices of
triangle ACE are A(1, 1), C(3, 3) and E(4, 1). What is the ratio
of the area of triangle ACE to the area of rectangle ABDE?
Express your answer as a common fraction.
B
A
3. ________________
C D
E
4. The European equivalent of 8 12 ” by 11” paper is called
A4 paper, and its dimensions are 0.21 meters by 0.297 meters.
What is the greatest total area, in square meters, that can be
covered by 21 sheets of rectangular A4 paper? Express your
answer as a decimal to the nearest tenth.
sq meters
4. ________________
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round
MATHCOUNTS
2006
Chapter Competition
Target Round
Problems 5 and 6
Name
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
Total Correct
Scorer’s Initials
Founding Sponsors
National Sponsors
National Society of
Professional Engineers
ADC Foundation
Raytheon Company
National Council of
Teachers of Mathematics
General Motors Foundation
Shell Oil Company
Lockheed Martin
Texas Instruments Incorporated
National Aeronautics and
Space Administration
3M Foundation
CNA Foundation
Northrop Grumman Foundation
Xerox Corporation
Copyright MATHCOUNTS, Inc. 2005. All rights reserved.
®
5. The arithmetic mean (or average) of A, B and C is 10. The
value of A is six less than the value of B, and the value of C is
three more than the value of B. What is the value of C?
5. ________________
6. A value is assigned to each letter of the alphabet such that
A = 1, B = 2, C = 3,..., Z = 26. A nine-digit code is then created
for each letter using the prime factorization of its assigned
value. The ﬁrst digit of a letter’s code is the number of times 2
is used as a factor, the second digit is the number of times 3 is
used as a factor, the third digit is the number of times 5 is used
as a factor, and so on. For example, since N is the 14th letter
of the alphabet and N = 14 = 21 × 71, the code for the letter N
is 100100000 with 1s in the ﬁrst and fourth positions because
its prime factorization has one 2 (the ﬁrst prime number) and
one 7 (the fourth prime number). What 6-letter word does the
following sequence of six codes represent? The ﬁrst row is the
code for the ﬁrst letter of the word, the second row is the code
for the second letter of the word, and so on.
6. ________________
0
0
0
0
2
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round
MATHCOUNTS
2006
Chapter Competition
Target Round
Problems 7 and 8
Name
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
Total Correct
Scorer’s Initials
Founding Sponsors
National Sponsors
National Society of
Professional Engineers
ADC Foundation
Raytheon Company
National Council of
Teachers of Mathematics
General Motors Foundation
Shell Oil Company
Lockheed Martin
Texas Instruments Incorporated
National Aeronautics and
Space Administration
3M Foundation
CNA Foundation
Northrop Grumman Foundation
Xerox Corporation
Copyright MATHCOUNTS, Inc. 2005. All rights reserved.
®
7. On a particular quiz, there are 15 easy questions and 15 hard
questions. The easy questions are worth 4 points each, and
the hard questions are worth 10 points each. No partial credit
is possible. Sam earns a total of 92 points on the quiz. What
is the greatest number of hard questions that he could have
answered correctly?
hard questions
7. ________________
8. Eli throws ﬁve darts at a circular target, and each one lands
within one of the four regions. The point-value of a dart
landing in each region is indicated. What is the least score
greater than ﬁve points that is not possible when the pointvalues of the ﬁve darts are added together?
points
8. ________________
6
4
2
1
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Target Round
MATHCOUNTS
®
2006
Chapter Competition
Team Round
Problems 1–10
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This section of the competition consists of ten problems
which the team has 20 minutes to complete. Team members
may work together in any way to solve the problems. Team
members may talk during this section of the competition. This
round assumes the use of calculators, and calculations may
also be done on scratch paper, but no other aids are allowed.
All answers must be complete, legible and simpliﬁed to lowest
terms. The team captain must record the team’s ofﬁcial
answers on his/her own problem sheet, which is the only sheet
that will be scored. If the team completes the problems before
time is called, use the remaining time to check your answers.
Team
Members
, Captain
Total Correct
Scorer’s Initials
Founding Sponsors
National Sponsors
National Society of
Professional Engineers
ADC Foundation
Raytheon Company
National Council of
Teachers of Mathematics
General Motors Foundation
Shell Oil Company
Lockheed Martin
Texas Instruments Incorporated
National Aeronautics and
Space Administration
3M Foundation
CNA Foundation
Northrop Grumman Foundation
Xerox Corporation
Copyright MATHCOUNTS, Inc. 2005. All rights reserved.
$
1. ________________
2. A ball bounces back up 23 of the height from which it falls. If
the ball is dropped from a height of 243 cm, after how many
bounces does the ball ﬁrst rise less than 30 cm?
bounces
2. ________________
450
3. The graph shows the maximum
400
speed and maximum height of
the eight fastest roller coasters
350
in the world in 2003. In 2003,
300
the Dodonpa roller coaster had
250
a maximum speed of
200
106.9 miles per hour, making
150
it one of the eight fastest roller
80 90 100 110 120
coasters. In 2003, how many
Speed (mph)
roller coasters had a maximum
speed that was faster than the Dodonpa roller coaster?
roller coasters
3. ________________
4. When Marika bought her house she paid 80% of the purchase
price of the house with a loan. She paid the remaining $49,400
of the purchase price with her savings. What was the purchase
price of her house?
$
4. ________________
5. One interior angle of a convex polygon is 160 degrees. The
rest of the interior angles of the polygon are each 112 degrees.
How many sides does the polygon have?
sides
5. ________________
6. A 25-passenger bus rents for $110, and a 40-passenger bus
rents for $170. What is the minimum cost for renting enough
buses for a school trip with 475 passengers?
$
6. ________________
Height (feet)
1. In 1992, a scoop of gelato could be purchased in Italy for
1200 lire. The same gelato would have cost $1.50 in the U.S.
At the equivalent exchange rate between the lire and the dollar,
how many dollars would be equivalent to 1,000,000 lire?
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Team Round
7. Arpan and Tomika want to place straw in the ﬂower beds
surrounding their house. In the ﬁgure below, every angle
is a right angle,
and the shaded
62’
regions represent the
20’
ﬂower beds. The
4’
20’
dimensions shown
6’
6’
6’
are measurements for
the house and do not
include the ﬂower beds. The width of the ﬂower beds is
2 feet. One bale of straw, costing $2.75, will cover 9 square
feet of ground. With 6% sales tax, how much will it cost them
to cover the ﬂower beds with straw, if only whole bales of
straw may be purchased?
$
7. ________________
8. A “value meal” consists of exactly one selection from the
entree menu, the drink menu and the dessert menu below. The
price of any value meal is calculated by subtracting
20 cents from the sum of the prices of the three individual
items. For example, the least expensive value meal costs
$2.45. If a customer orders exactly one of each of the three
most expensive value meal combinations, what is the total cost
of the customer’s three value meals?
$
8. ________________
Entree
Hot dog - $1.25
Hamburger - $1.65
Chicken - $1.80
Pizza - $2.25
Drink
Coffee - $0.75
Soda - $1.05
Juice- $1.25
Dessert
Cookie - $0.65
Pudding - $1.30
9. A 6 by 6 grid of 36 unit squares is completely covered in
T-shaped pieces consisting of four unit squares. The T-shaped
pieces do not overlap, but pieces may extend over the side of
the 6 by 6 grid. What is the fewest number of
T-shaped pieces required to cover the 6 by 6 grid?
pieces
9. ________________
10. Three friends share a full bag of jellybeans. Mike took 13 of
the jellybeans in the full bag, Zac took 12 of the jellybeans in
the full bag and Kary took what was left. Mike ate 12 of his
jellybeans, Zac ate 13 of his jellybeans and Kary ate all of hers.
If Mike and Zac now have a total of 45 jellybeans together,
how many jellybeans did Kary eat?
jellybeans
10. ________________
Copyright MATHCOUNTS, Inc. 2005. All rights reserved. 2006 Chapter Team Round